Preface

This book deals with combination-based approach for permutation hypotheses testing in several complex problems frequently encountered in practice. It also deals with a wide range of difficult applications in easy-to-check conditions. The key underlying idea, on which most part of testing solutions in multidimensional settings is based, is the non-parametric combination (NPC) of a set of dependent partial tests according to Roy's Union-Intersection idea. This methodology assumes that a testing problem is properly broken down into a set of simpler sub-problems, each provided with a proper permutation solution, and that these sub-problems can be jointly analysed in order to maintain underlying unknown dependence relations.

The first four chapters are devoted to updated theory on univariate and multivariate permutation tests. The remaining chapters present real case studies (mainly observational studies) along with recent developments in permutation solutions. Observational studies have enjoyed increasing popularity in recent years for several reasons, including low costs and availability of large data sets, but they differ from experiments because there is no control of the assignment of treatments to subjects. In observational studies the experimenter's main interest is usually to discover an association among variables of interest, possibly indicating one or more causal effects. The robustness of the nonparametric methodology against departures from normality and random sampling are much more relevant in observational studies than in controlled trials. Hence, in this context, the NPC method is particularly suitable. Moreover, given that the NPC method is conditional on a set of sufficient statistics, it shows good general power behaviour, and the Fisher, Liptak-Stouffer or direct combining functions often have power functions which are quite close to the best parametric counterparts, when the latter are applicable, even for moderate sample sizes. Thus NPC tests are relatively efficient and much less demanding in terms of underlying assumptions with respect to parametric competitors and to traditional distribution-free methods based on ranks, which are generally not conditional on sufficient statistics and so rarely present better unconditional power behaviour. One major feature of the NPC with dependent tests, provided that the permutation principle applies, is that we must pay attention to a set of partial tests, each appropriate for the related sub-hypotheses, because the underlying dependence relation structure is nonparametrically and implicitly captured by the combining procedure. In particular the researcher is not explicitly required to specify the dependence structure on response variables. This aspect is of great importance particularly for non-normal and categorical variables in which dependence relations are generally too difficult to define, and even when well-defined, are hard to cope with. Furthermore, in the presence of a stratification variable, NPC through a multi-phase procedure allows for quite flexible solutions. For instance, we can firstly combine partial tests with respect to variables within each stratum and then combine the combined tests with respect to strata. Alternatively, we can firstly combine partial tests related to each variable with respect to strata and then combine the combined tests with respect to variables. Moreover, once a global inference is found significant, while controlling for multiplicity it is possible to recover which partial inferences are mostly responsible of that result.

Although dealing with essentially the same methodology contained in Pesarin (2001), almost all material included in this book is new, specifically with reference to underlying theory and case studies.

Chapter 1 contains an introduction to general aspects and principles concerning the permutation, i.e. conditional, approach. The main emphasis is on principles of conditionality, sufficiency, similarity, relationships between conditional and unconditional inferences, why and when conditioning may be necessary, why permutation approach results from both conditioning with respect to the data set and exchangeability of data in the null hypothesis etc. Moreover permutation techniques are discussed along with computational aspects. Then, basic notation is introduced. Through a heuristic discussion of simple examples on univariate problems with paired data, two-sample and multi-sample (one-way ANOVA) designs, the practice of permutation testing is introduced. Moreover, discussions on conditional Monte Carlo methods (CMC) for estimating the distribution of a test statistic and some comparisons with parametric and nonparametric counterparts are also presented.

Chapters 2 and 3 formally present: the theory of permutation tests for unidimensional one-sample and multisample problems; proof and related properties of conditional and unconditional unbiasedness; definition and derivation of conditional and unconditional power functions; emphasis is given on the weak consistency of permutation tests without requiring existence of second moments; confidence intervals for treatment effect the extension of conditional inferences to unconditional counterparts; a brief discussion on optimal permutation tests and of permutation central limit theorem.

Chapter 4 presents the multivariate permutation testing with the NPC methodology. The presentation includes a discussion on assumptions, properties, sufficient conditions for a complete theory of NPC of dependent tests, and practical suggestions for making a reasonable selection of the combining function to be used when dealing with practical problems. Also discussed are concept of finite-sample consistency, especially useful when the number of observed variables in each subject exceeds that of subjects in the study; the multi-aspect approach; separate testing for two-sided alternatives; testing for multi-sided alternatives; taking for equivalence and non-inferiority the Behrens-Fisher problem, etc.

Chapter 5 deals with multiple comparisons and multiple testing issues. A brief overview of multiple comparison procedures (MCPs) is presented. The main focus is on closed testing procedures for multiple comparisons and multiple testing. Some hints are also given with reference to weighted methods for controlling FWE and FDR, adjustment of stepwise -values, and optimal subset procedures.

Chapter 6 concerns multivariate permutation approaches for categorical data. A natural multivariate extension of McNemar's test is presented along with multivariate goodness-of-fit test for ordinal variables, MANOVA test with nominal categorical and stochastic ordering issue in presence of multivariate categorical ordinal variables. A permutation approach to test allelic association and genotype-specific effects in the genetic study of a disease is also discussed. An application problem concerning how to establish if the distribution of a categorical variable is more heterogeneous (less homogeneous) in one population than in another is presented as well.

Chapter 7 discusses some quite particular problems with repeated measurements and/or missing data. Carry-over effects in repeated measures designs, modelling and inferential issues are treated extensively. Moreover testing hypothesis problems for repeated measurements and missing data are examined. Remaining part of the chapter is devoted to permutation testing solutions with missing data.

Chapter 8 refers to permutation approaches for hypothesis testing when a multivariate monotonic stochastic ordering is present (both with continuous and/or categorical variables). Umbrella testing problems are also presented. Moreover, two applications are also discussed: one concerning the comparison of cancer growth patterns in laboratory animals and the other referring to a Functional Observational Battery (FOB) study designed to measure the neurotoxicity of perchloroethylene, a solvent used in dry cleaning (Moser, 1989; McDaniel and Moser, 1993).

Chapter 9 concerns permutation methods for problems of hypothesis testing in the framework of survival analysis. Two applications in the medical field are presented and discussed.

Chapter 10 deals with statistical shape analysis. Most of the inferential methods known in the shape analysis literature are parametric in nature. They are based on quite stringent assumptions, such as the equality of covariance matrices, the independency of variation within and among landmarks or the multinormality of the model describing landmarks. But, as known, the assumption of equal covariance matrices may be unreasonable in certain applications, the multinormal model in the tangent space may be doubted and sometimes there are fewer individuals than landmarks, implying over-dimensioned spaces and loss of power. On the strength of these considerations, an extension of NPC methodology to shape analysis is suggested. Focussing on the two independent sample case, through an exhaustive comparative simulation study, the behaviour of traditional tests along with nonparametric permutation tests using multi-aspect procedures and domain combinations as well is evaluated. The case of heterogeneous and dependent variation at each landmark is also analyzed, along with the effect of superimposition on the power of NPC tests.

Chapter 11 presents two interesting real case studies in ophthalmology, concerning complex repeated measures problems. For each data set, different analyses have been proposed in order to highlight peculiar aspects of the data structure itself. In this way we enable the reader to choose most appropriate...